Cyclotomic algebras

1974 ◽  
pp. 14-19
Author(s):  
Toshihiko Yamada
Keyword(s):  
Cyclotomic Algebras

The Schur Subgroup of a p-Adic Field

1979 ◽  
Vol 31 (2) ◽  
pp. 300-303
Author(s):  
Eugene Spiegel ◽  
Allan Trojan
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Group Algebras ◽  
Brauer Group ◽  
Ramification Index ◽  
Simple Component ◽  
Image Position ◽  
Cyclotomic Algebras ◽  
A Finite Group ◽  
Tame Ramification

Let K be a field. The Schur subgroup, S(K), of the Brauer group, B(K), consists of all classes [△] in B(K) some representative of which is a simple component of one of the semi-simple group algebras, KG, where G is a finite group such that char K ∤ G. Yamada ([11], p. 46) has characterized S(K) for all finite extensions of the p-adic number field, Qp. If p is odd, [△] ∈ S(K) if and only ifwhere c is the tame ramification index of k/Qp, k the maximal cyclotomic subfield of K, and s = ((p – 1)/c, [K : k]). invp △ is the Hasse invariant. Yamada showed this by proving first that S(K) is the group of classes containing cyclotomic algebras and then determining the invariants of such algebras.

Gyclotomic Division Algebras

1981 ◽  
Vol 33 (5) ◽  
pp. 1074-1084 ◽  
Author(s):  
R. A. Mollin
Keyword(s):  
Finite Group ◽  
Number Field ◽  
Sufficient Conditions ◽  
Algebraic Number Field ◽  
Equivalence Classes ◽  
Necessary And Sufficient Conditions ◽  
Brauer Group ◽  
Division Algebras ◽  
Cyclotomic Algebras ◽  
Necessary And Sufficient

Let K be a field of characteristic zero. The Schur subgroup S(K) of Brauer group B(K) consists of those equivalence classes [A] which contain an algebra which is isomorphic to a simple summand of the group algebra KG for some finite group G. It is well known that the classes in S(K) are represented by cyclotomic algebras, (see [16]). However it is not necessarily the case that the division algebra representatives of these classes are themselves cyclotomic. The main result of this paper is to provide necessary and sufficient conditions for the latter to occur when K is any algebraic number field.Next we provide necessary and sufficient conditions for the Schur group of a local field to be induced from the Schur group of an arbitrary subfield. We obtain a corollary from this result which links it to the main result. Finally we link the concept of the stufe of a number field to the existence of certain quaternion division algebras in S(K).

scholarly journals The gap between the Schur group and the subgroup generated by cyclic cyclotomic algebras

2010 ◽  
Vol 176 (1) ◽  
pp. 401-417
Author(s):  
Allen Herman ◽  
Gabriela Olteanu ◽  
Ángel del Río
Keyword(s):  
Cyclotomic Algebras

scholarly journals On the generic degrees of cyclotomic algebras

2000 ◽  
Vol 4 (14) ◽  
pp. 342-369 ◽  
Author(s):  
Gunter Malle
Keyword(s):  
Cyclotomic Algebras

scholarly journals Ring Isomorphism of Cyclic Cyclotomic Algebras

2009 ◽  
Vol 12 (2-5) ◽  
pp. 365-370 ◽  
Author(s):  
Allen Herman ◽  
Gabriela Olteanu ◽  
Ángel del Río
Keyword(s):  
Ring Isomorphism ◽  
Cyclotomic Algebras

scholarly journals Cyclotomic algebras over a $2$-adic field

1973 ◽  
Vol 49 (6) ◽  
pp. 438-442 ◽  
Author(s):  
Toshihiko Yamada
Keyword(s):  
Cyclotomic Algebras
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